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# STATS 120C University of California Linear Regression Model Question

See the attached file for the correct version since it doesn’t show right on the website

1. Consider the following simple linear regression modelYi =?0+?1Xi+?i

with ?i independent normall-distributed errors with E [?i] = 0 and Var[?i] =?2, for i = 1,…,n (e.g. ?i ? N(0,?2)). A common use of the regression model is to estimate the value of the outcome Y conditional upon a given valueofX. AnaturalestimateofE[Y|X =Xh]isgivenbyYË†h:

YË†h = ?Ë†0+?Ë†1Xh
= Y? ? ?Ë† 1 X? + ?Ë† 1 X h= Y?+?Ë†1(Xh?X?)

(a)  FindtheexpectationofYË†h.IsYË†hanunbiasedestimatorofE[Y|X=Xh]? (b)  Show that the variance of YË†h is given by

n ? ni = 1 ( X i ? X? ) 2(Hint:youwillneedtousethefactthatCovY?,?Ë† =0.

(c) WhatisthedistributionofYË†hundertheassumptionsofthemodelgiven above?

(d) Suppose we estimate ?2 by s2 = RSS , derive the distribution ofn?2

1 (X?X?)2 ?2 + h

YË† h ? E [ Y | X = X h ]1 (Xh?X?)2

1

s2 n + n ? 2?i=1 (Xi ?X )

(Hint: you can use the fact that RSS ? ?2 that we mentioned in class

?2 n?2
(e) Usingyourresultffrom(d),howyouestimatea95%CIforE[Y|X=Xh]?

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